Inequalities that Collectively Completely Characterize the Catalytic Majorization Relation

For probability vectors x and y, the catalytic majorization relation x precT y is defined to hold when there exists a probability vector z such that x otimes z is majorized by y otimes z. In this paper, an infinite family of functions is given such that, subject to some trivial restrictions, x precT y if and only if fᵣ (x) < fᵣ (y) for all functions fᵣ in the family. An outline of a proof of this result is provided. The catalytic majorization relation is known to provide a determination of which transformations of jointly held pure quantum states are possible using local operations and classical communication when an additional jointly held state may be specified to facilitate the transformation without being consumed.